Method for sending classical data in quantum information processing systems and corresponding system

ABSTRACT

Method for sending first data as quantum information in qubits (Iφ&gt;) and classical second data (Si) over a quantum channel (12; 12a; 12b), in particular in quantum information communication systems (10; 10a; 10b), which includes applying QECC encoding (111) to said qubits ((Iφ&gt;) obtaining quantum information codewords (Iψ&gt;), wherein said method (200; 300) includes applying (210) intentional errors (Pi) with error syndromes (Si) representing said second classical data to said quantum information code-words ((Iψ&gt;) obtaining quantum information codewords with intentional errors (P1) applied upon (PiIψi&gt;), and transmitting (220) from a transmitting side (11; 11a; 11b) said quantum information codewords with intentional errors applied upon (PiIψi&gt;) over said quantum channel (12; 12a) which outputs received codewords (PiIψi&gt;;EiPiIψi&gt;) at a receiving side (13; 13b), computing (230; 330) error syndromes (Si,Ri) from said received codewords (PiIψi&gt;;EiPiIψi&gt;), performing a QECC error correction operation (250; 350) on said received codewords (PiIψi&gt;;EiPiIψi&gt;) by applying a correction operator (Pi+; Pi+Ei+) obtained at least by said computed syndromes (Si; Ri) to obtain corrected codewords (Iψi&gt;), outputting (260; 360) said corrected codewords (Iψi&gt;) and said computed syndromes (Si).

TECHNICAL FIELD

The present description relates to techniques for sending first data asquantum information in qubits and second classical data in quantuminformation processing systems over a quantum channel, which includesapplying QECC encoding to said qubits obtaining quantum informationcodewords.

The techniques here described refer to classical data which representscontrol data for instance for the quantum information network orprocessing system, or additional data which represent supplementaryinformation, or data which represent synchronization data for thequantum information network or processing system.

TECHNOLOGICAL BACKGROUND

Quantum information processing by systems including quantum computersand quantum networks have been known since some year. Despite thepotential advantages in exploiting the peculiarities of quantummechanics to process information, there are still several problems tosolve in the path towards large-scale quantum computers and quantumnetworks.

One aspect is that the management of such a network will require toexchange control data in addition to the user data. Nodes should be ableto identify each packet within a stream of qubits (synchronization), andalso to write and read management and control information attached tothe qubit stream. For instance, in classical networks adopting theInternet Protocol (IP) each packet contains the source and destinationaddresses, as well as a hop counter, used and updated by routers. In thefollowing we call “control data” all information besides the user data.In general it is referred to classical data as data which can bedescribed by bits, while quantum information data are carried by quantumstates. In classical networks the control data can be transferred on thesame physical channel used to carry the user information (in-bandcontrol). For example, a fixed pattern of bits (sync word) can beinserted in a packet for frame synchronization. A receiving node readsthe bits, for instance by a sliding correlator, until it finds the syncword, indicating the boundaries of a packet. After synchronization, theaddress of the destination contained in each packet is read and used toforward the packet toward the destination.

However, inserting qubits as control data is not always a viableapproach for quantum networks, since in general measuring destroysquantum state superposition. For this reason, several studies assumethat quantum networks will need out-of-band control and signaling, sinceany attempt to read and process control information carried in thequantum channel will destroy its content (see e.g. Quantum Networks forOpen Science Workshop. Rockville, Md., USA: Office of Science UDepartment of Energy, 2018, section 2.5). For example, synchronizationpatterns of qubits cannot be just embedded in the quantum stream, likein classical networks, as reading the qubit stream until their positionis found would destroy the superposition on the user information qubits.

A possible way to solve this problem is to introduce auxiliaryorthogonal states used just for synchronization purposes. For instance,one might build a quantum system with qutrits (Hilbert space ofdimension three) instead of the usual qubits, where the orthogonalstates |0

, |1

are used as the basis for information, and an additional orthogonalstate |2

is used for synchronization purposes, as for instance discussed in Y.Fujiwara, “Parsing a sequence of qubits”, IEEE Transactions onInformation Theory, vol. 59, no. 10, pp. 6796-6806, 2013. Also, placingpatterns of states |2

inserted in different positions along the quantum stream can be used tocarry simple metadata. It must be noted that changing the metadata wouldrequire to change the pattern of insertions of the |2

's, so that the classical information is practically of read-only type.Besides this limitation, the main difficulty here is related to the needof working with qutrits instead of qubits, with an impact on the overallgeneral system architecture.

Thus, there is the need of an improved solution which allows sendingcontrol, or additional, data along with the quantum stream.

OBJECT AND SUMMARY

An object of one or more embodiments is to contribute in providing suchan improved solution.

Specifically, an object of the invention is to provide a method thatallows operating with qubits without needing out-of-band control andsignaling.

According to one or more embodiments, that objective can be achieved bymeans of a method having the features set forth in the claims thatfollow. Embodiments moreover concerns a related quantum informationtransmission system.

As mentioned in the foregoing, the present disclosure provides solutionsregarding a method for sending first data as quantum information inqubits and classical second data over a quantum channel, in particularin quantum information communication systems, which includes applyingQECC encoding to said qubits obtaining quantum information codewords,

-   -   wherein said method includes    -   applying intentional errors having error syndromes representing        said second classical data to said quantum information codewords        obtaining quantum information codewords with intentional errors        applied upon, and    -   transmitting from a transmitting side said quantum information        codewords with intentional errors applied upon over said quantum        channel which outputs received codewords at a receiving side,    -   computing error syndromes from said received codewords,    -   performing a QECC error correction operation on said received        codewords by applying a correction operator obtained at least by        said computed syndromes to obtain corrected codewords,    -   outputting said corrected codewords and said computed syndromes.

In variant embodiments, the method includes encoding information datawith a classical error code correction encoder to obtain encoded errorsyndromes which are applied as said error syndromes to the quantuminformation codewords,

-   -   said computing error syndromes from said received codewords        including        -   a step of computing channel affected syndromes from the            received codewords and a step of classical syndrome error            correction on said channel affected syndromes to obtain            classical corrected error syndromes,    -   said applying a correction operator obtained at least by said        computed syndromes to obtain corrected codewords includes    -   obtaining said correction operator by    -   performing an operation of computation of the intentional error        and of the channel error to which is associated the syndrome on        the basis of said channel affected syndromes and said classical        corrected error syndromes,    -   using said computed intentional error and channel error to        obtain said correction operator, in particular as inverse of the        computed intentional error and channel error.

In variant embodiments, the method includes that said applying acorrection operator obtained at least by said computed syndromes toobtain corrected codewords includes obtaining the intentional error fromsaid computed syndromes and computing the correction on the basis ofsaid intentional error, in particular as inverse of the intentionalerror.

In variant embodiments, the method includes applying intentional errorshaving error syndromes representing said second classical data to saidquantum information codewords obtaining quantum information codewordswith intentional errors applied upon includes introducing in the quantuminformation codewords intentional errors determined by correspondingsyndromes.

In variant embodiments, the method includes that said second classicaldata represent communication control data.

In variant embodiments, the method includes that said second classicaldata represent a synchronization word which is attached to selectedcodewords.

The method here described allows, for any quantum communication systememploying quantum error correcting codes (QECCs), to read and writeclassical information on top of quantum information. Specifically, themethod defines a communication protocol to send a sequence of classicalbits superimposed to qubits protected by QECCs by the introduction ofintentional errors on the qubits, so that the classical information isconstituted by the error syndrome sequence.

The present disclosure provides also solutions regarding a quantuminformation transmission system configured to send first data as quantuminformation in qubits and classical second data over a quantum channel,comprising a quantum information transmission module which includes aQECC encoder configured to apply QECC encoding to said qubits obtainingquantum information codewords,

-   -   wherein said quantum information transmission module is        configured to    -   apply intentional errors having error syndromes representing        said second classical data to said quantum information codewords        obtaining quantum information codewords with intentional errors        applied upon, and    -   transmit from a transmitting side said quantum information        codewords with intentional errors applied upon over said quantum        channel which outputs received codewords at a receiver module        comprised in said system,    -   said receiver module being configured to    -   compute error syndromes from said received codewords,    -   perform a QECC error correction operation on said received        codewords by applying a correction operator obtained at least by        said computed syndromes to obtain corrected codewords,    -   output said corrected codewords and said extracted syndromes.

In variant embodiments, said transmitter module includes a classicalerror code correction encoder configured to encode information data toobtain encoded error syndromes which are applied as said error syndromesto the quantum information codewords,

-   -   said receiver module configured to compute error syndromes from        said received codewords including one or more modules configured        to        -   compute channel affected syndromes from the received            codewords, and        -   perform a classical syndrome error correction on said            channel affected syndromes to obtain classical corrected            error syndromes,    -   an application of a correction operator obtained at least by        said computed syndromes to obtain corrected codewords including    -   obtaining said correction operator by        -   performing an operation of computation of the intentional            error and of the channel error on the basis of said channel            affected syndromes and said classical corrected error            syndromes,        -   using said computed intentional error and channel error to            obtain said correction operator, in particular as inverse of            the computed intentional error and channel error.

In variant embodiments, the receiver module is configured to apply acorrection operator obtained at least by said computed syndromes toobtain corrected codewords includes obtaining the intentional error fromsaid computed syndromes and computing the correction on the basis ofsaid intentional error, in particular as inverse of the intentionalerror.

The claims are an integral part of the technical teaching providedherein with reference to the embodiments.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Various embodiments will now be described, purely by way of example,with reference to the annexed drawings, wherein:

FIG. 1 is a block schematic of a communication system according to theprior art;

FIG. 2 is a block schematics of a channel according to the presentsolution;

FIG. 3 is a block schematic of a first embodiment of a communicationsystem implementing the method here described;

FIG. 4 is a block schematic of a second embodiment of a communicationsystem implementing the method here described;

FIGS. 5 to 7 represents quantum circuits according to the prior artwhich can be used to implement the method here described.

FIG. 8 represents a diagram flow of an embodiment of the method heredescribed;

FIG. 9 represents a diagram flow of a further embodiment of the methodhere described.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

In the ensuing description numerous specific details are illustrated inorder to enable maximum understanding of the embodiments provided by wayof example. The embodiments may be implemented with or without specificdetails, or else with other processes, components, materials, etc. Inother circumstances, structures, materials, or operations that are wellknown are not shown or described in detail so that various aspects ofthe embodiments will not be obscured. Reference, in the course of thepresent description, to “an embodiment” or “one embodiment” indicatesthat a particular feature, structure, or characteristic described inconnection with the embodiment is comprised in at least one embodiment.Hence, phrases such as “in an embodiment” or “in one embodiment” thatmay be present in various points of the present description do notnecessarily refer to one and the same embodiment. Moreover, theparticular features, structures, or characteristics may be combined inany convenient way in one or more embodiments.

The terms and references are provided herein merely for convenience ofthe reader and do not define the sphere of protection or the scope ofthe embodiments.

The solution here described aims to overcome these drawbacks. The methodhere described which allows, for any quantum communication systememploying quantum error correcting codes (QECCs), to read and writeclassical information on top of quantum information. Specifically, themethod defines a communication protocol to send a sequence of classicalbits superimposed to qubits protected by QECCs by the introduction ofintentional errors on the qubits, so that the classical information isconstituted by the error syndrome sequence.

The method here described applies to quantum systems employing QuantumError Correction by encoding qubits with Quantum Error Correcting Codes,QECC. Thus, here it is defined the notation and the main elements ofQECC. Further information can be found for instance in the publications:

-   D. Gottesman, “An introduction to quantum error correction and    fault-tolerant quantum computation,” in Quantum information science    and its contributions to mathematics, Proceedings of Symposia in    Applied Mathematics, vol. 68, 2009, pp. 13-58;-   M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum    Information. Cambridge University Press, 2010;-   Z. Babar, D. Chandra, H. V. Nguyen, P. Botsinis, D. Alanis, S. X.    Ng, and L. Hanzo, “Duality of quantum and classical error correction    codes: Design principles and examples”, IEEE Communications Surveys    Tutorials, vol. 21, no. 1, pp. 970-1010, Firstquarter 2019.

Quantum information is here defined as information carried by qubits.

A qubit is an element of the 2-dimensional Hilbert space,

². The standard computational basis is denoted by |0

, |1

. An n-tuple of qubits (n qubits) is an element of the 2n-dimensionalHilbert space,

^(2{circumflex over ( )}n), with standard computational basis composedby all possible |i₁

⊕|i₂

⊕. . . ⊕|i_(n)

with i_(j)∈{0,1 }, 1≤j≤n.

For a ∈{0,1}the Pauli operators are denoted as I, X, Z, Y and defined asI|α

=|α

, X|α

=|α⊕1

, Z|α

=(1)^(α)|α

, Y|α

=i(−1)^(α)|α⊕1

. These operators either commute or anticommute. The Pauli group G_(n)on n qubits is generated by all n-fold tensor products of these fouroperators together with the factors ±1 and ±i. Two operators in groupG_(n) commute if and only if there is an even number of places wherethey have different Pauli matrices, neither of which is the identity I.

The solution here described substantially provides, applying intentionalerrors upon quantum information codewords, i.e. piggybacking viaintentional errors for both noiseless and noisy quantum channels.

The solution here described adds classical information, i.e. seconddata, preferably control data, on top of quantum informationrepresenting first data. For example, control data may includeannotating the qubits to describe what they represent, who producedthem, etc.; more important, it is allowed reading and rewriting thisannotation without destroying the quantum information. Another examplemay be related to the possibility to have a quantum network (QN), wherenodes exchange quantum information organized in packets.

In FIG. 1 it is shown a quantum communication system block schematicsrepresenting a quantum communication system 10 employing Quantum ErrorCorrection. Such quantum communication system 10 operates between twonodes, e.g. a transmitting side and a receiving side, using QECC to copewith impairments of a quantum channel 12. Such quantum communicationsystem 10, as the systems 10 a, 10 b described in the following, may bepart of a larger communication system, a quantum network or a quantumprocessing system. The embodiments shown may apply to all QECC schemes,and for the sake of simplicity it is here considered the case of blockcodes, the generalization to other codes being straightforward. Thequantum communication system 10 includes on a transmission side of thequantum channel 12 a transmission module 11 configured to transmitqubits |φ

over the quantum channel 12 to a receiver module 13. The transmissionmodule 11 includes a quantum encoder 111 which encodes first datarepresented by data (logical) k qubits |φ

into an n qubits q-codeword |ψ

according to a stabilizer code C, generated by n−k independent operatorsg_(j)∈G_(n),j=1, 2 . . . n−k. The n qubits q-codeword |ψ

is sent over the quantum channel 12.

In the following, q-codeword and c-codeword indicate if it is referredto quantum information or classical information codewords, respectively.

The codewords are eigenvectors with eigenvalues +1 for all thegenerators, i.e. ∀|ψ

∈

it is:

g _(j) |ψ

=+|ψ

, j=1, 2 . . . n−k   (1)

while for ∀|ψ

∉

there exist at least a j such that g_(j)|ψ

=−|ψ

. It is said that a codeword is stabilized by all generators g_(j).

A codeword |ψ

∈

may be affected by a channel error represented by the operator E∈G_(n),i.e. the error of quantum channel 12. For any generator g_(j) the errorE either commutes or anticommutes. For error correction, the received nqubit E|ψ

, i.e. the output of quantum channel 12, is measured according to allthe generators g₁,g₂, . . . ,g_(n-k). in a syndrome computation block131 obtaining a quantum error syndrome

(E)=s₁, . . . , s_(n−k), with syndrome element s_(j)=+1/−1 if error Ecommutes/anticommutes with generator g_(j). Note that, due to eq. (1),the quantum error syndrome

(E) depends on error E and not on the particular q-codeword |ψ

. Measuring the syndrome

(E) should not change the quantum state, which remains the received nqubit or codeword E|ψ

, which corresponds to the output of quantum channel 12 in FIG. 1. Thereare m=2n−k possible distinct syndromes S⁽¹⁾, S⁽²⁾, . . . S^((m)). It ishere indicated by S⁽¹⁾=(+1, +1, . . . +1) the syndrome of the operatorsE (including the identity I, i.e., the no-errors operator) such that thereceived qubit E|ψ

is still a valid q-codeword. Also, it is denoted by Ω^((i)) the set ofchannel errors on the n qubits producing the syndrome S^((i)), and byQ^((i))∈G_(n) the error in the set Ω⁽¹⁾ of channel errors which isassumed for error correction by the QECC decoding block 132. It isunderlined that, for degenerate quantum codes, correcting according toerror Q^((i)) may also work to correct other errors in the set Ω^((i))which act the same way on the codewords. The set of correctable errorsis identified by {Q^((i)), Q⁽²⁾, . . . , Q^((m))}, and their equivalentsfor degenerate codes.

The measured syndrome is used to estimate the error, i.e. the block 131supplies the measured syndrome to Quantum Error Correcting Codes (QECC)decoder 132. More precisely, if the measured syndrome, measured by block131, is S^((i)), then the QECC decoder 132 assumes the error Q^((i)) (orone equivalent to it for degenerate codes) occurred. Once the error hasbeen detected, the inverse error operator (Q^((i)))^(†) can be appliedto recover the state to a valid codeword.

A simple example of QECC encoding is that of a repetition code, whichmaps a data qubit α|0

+β|1

into a q-codeword α|000

+β|111

. In FIG. 5 it is represented a quantum circuit embodiment of the QECCencoder 111 performing such encoding. As generators it is possible tochoose g_(i)=Z⊗Z⊗I and g₂=I⊗Z⊗Z. Measuring a q-codeword will thereforeproduce the syndrome S⁽¹⁾=(+1, +1). The code can correct any singlebit-flip error occurring on a q-codeword: if the first qubit is flipped,the received 3 qubits status is α|100

+β|011, and the measured syndrome will be S⁽²⁾=(−1,+1). It is underlinedthat, after the measurements, the state of the received 3 qubits remainsunchanged. The syndrome S⁽²⁾ is due to one bit-flip error on the firstqubit, or to the bit-flip of the last two qubits.

Assuming the code is used for single errors, the QECC decoder 132 willjust apply Q⁽²⁾=X⊗I⊗I to the received 3 qubits in order to bit-flip thefirst qubit, recovering the error. Similarly, a bit-flip on the secondqubit will produce the syndrome S⁽³⁾=(−1, −1), and a bit-flip on thethird will produce the syndrome S⁽¹⁾=(+1, −1) .

Now it is described the method for sending first data as quantuminformation in qubits and second data, i.e. classical informationpreferably representing control data, in quantum information processingsystems over a quantum channel, which includes applying QECC encoding tosaid qubits obtaining quantum information codewords according to theinvention.

Given a (n,k) QECC encoding used, i.e. in block 111, to encode asequence |φ₀

, |φ₁

, |φ₂

, . . . of k data qubits producing a sequence of n qubits q-codewords|ψ₀

, |ψ₁

, |ψ₂

, . . . with |ψ

∈

, the method provides, instead of transmitting on the channel 12 thecodewords |ψ_(i)

as in FIG. 1 from the quantum encoder 111, previously inserting at thetransmitter side 11 deliberately and in a controlled way, bypiggybacking, some intentional errors P₀,P₁,P₂ . . . indicating withS₀,S_(i),S₂, . . . the corresponding error syndromes. It is chosen thatthe intentional errors P_(i) belong to the set of correctable errors,i.e. P_(i)∈{Q⁽¹⁾, Q⁽²⁾, . . . , Q^((m))} so that such intentional errorsP_(i) can be later corrected by the QECC correction block, or QECCdecoder, 132. At the receiver side 13 a, the block 132 which computesthe quantum error syndromes is configured to estimate a sequence ofmeasured syndromes errors Ŝ₀,Ŝ₁,Ŝ₂ . . .

In this way it is created, in piggyback on the quantum stream, an m-arydiscrete-input discrete-output classical channel, where the symbolalphabet, for both the input and the output of such classical channel isthe set of all possible syndromes {S⁽¹⁾, S⁽²⁾, . . . , S^((m))}, withm=2n−k. Thus the set of all possible syndromes {S⁽¹⁾, S⁽²⁾, . . . ,S^((m))} represents the set of symbols to represent the second classicaldata, in particular control data or sync words. This m-arydiscrete-input discrete-output classical channel is called here thepiggyback syndrome channel (PSC), indicated with 20 in the simple blockschematics of FIG. 2, to which error syndromes S₀, S_(i), S₂, are inputand from which measured syndromes Ŝ₀,Ŝ₁,Ŝ₂ . . . are output.

Now, the method here proposed is described in detail with reference tothe block schematics in figure of a communication system 10 a, whichincludes a transmitter module 11 a at the transmitter side whichtransmits quantum information over a noiseless quantum channel 12 a to areceiver module 13 at the receiver side. The case of a noisy channel istreated afterwards. The transmitter module 11 a includes the QECCencoder 111 generating from a i-th qubits |φ_(i)

, i being the index in a sequence of input qubits, a correspondingq-codeword |ψ_(i)

. A block 112 in the transmitter module 11 a represents a controllederror insertion block 112 which is configured to apply an intentionalerror P_(i)∈{Q⁽¹⁾, Q⁽²⁾, . . . , Q^((m))} to said i-th q-codeword |ψ_(i)

. The controlled error insertion block 112 is configured to apply,through application of intentional errors P_(i) to the i-th q-codeword|ψ_(i)

, error syndromes, representing second classical data with respect tofirst data represented by the input qubits |φ_(i)

, to the quantum information codewords |ψ_(i)

, obtaining quantum information codewords with intentional errorsapplied upon, i.e. piggybacked, P_(i)|ψ_(i)

. Piggybacking in the context of the solution here described meansattaching classic information to the q-codeword, by introducing in theq-codeword intentional errors, i.e. specifically applying an intentionalerror operator P_(i) to the ket representing the quantum state of theencoded q-codeword. The sequence of error syndromes S₀,S₁,S₂. . .represents the classical information that it is desired to piggyback onthe quantum stream. Since the noiseless quantum channel 12 a does notintroduce further errors, at the output of the channel are received nqubits corresponding to the quantum information codewords withintentional errors applied upon P_(i)|ψ_(i)

, which are sent to the syndrome computation block 131 for extractingthe syndrome. The measured syndrome measured at block 131 is Ŝ_(i)=

(P_(i))=S_(i).

It is underlined that in FIG. 3 lines transporting classical data areshown by a double line, while lines transporting quantum information arerepresented by a single line.

Therefore, the received sequence of measured syndromes, Ŝ₀,Ŝ₁,Ŝ₂ . . .reproduces the sequence of error syndromes S₀,S₁,S₂ . . . transmittedthrough the PSC channel 20, which in this case correspond to blocks 112,12 a, 131. Then, from the measured syndromes, the intentional quantumerrors P_(i) on the received n qubits, i.e. quantum informationcodewords with intentional errors applied upon, P_(i)|ψ_(i)

are sent to QECC correction block 132 in the receiver 13 to be correctedby applying {circumflex over (P)}_(i) ^(†), i.e. the inverse ofintentional quantum error operators P_(i), restoring the q-codewords|ψ_(i)

. Since the computation of syndromes at the receiver side, i.e. module13 a, does not destroy quantum superposition, with this syndrome-basedtransmission method it is obtained a noiseless discrete-inputdiscrete-output classical channel superimposed to the quantum stream.

This method can be used to annotate the quantum stream for several uses.For example, it is possible to add to a group of q-codewords adescription in the form of classical bits, which can be read andrewritten without altering the quantum information.

An example is illustrated in Table 1 herebelow.

TABLE 1 |ψ_(i )

. . . |ψ₁ 

|ψ₂ 

|ψ₃ 

|ψ₄ 

|ψ₅ 

|ψ₆ 

|ψ₇ 

. . . S_(i) . . . s⁽³⁾ s⁽¹⁾ s⁽²⁾ s⁽⁴⁾ s⁽⁴⁾ s⁽¹⁾ s⁽³⁾ . . . P_(i)|ψ_(i )

. . . Q⁽³⁾|ψ₁ 

Q⁽¹⁾|ψ₂ 

Q⁽²⁾|ψ₃ 

Q⁽⁴⁾|ψ₄ 

Q⁽⁴⁾|ψ₅ 

Q⁽¹⁾|ψ₆ 

Q⁽³⁾|ψ₇ 

. . . Ŝ_(i) . . . s⁽³⁾ s⁽¹⁾ s⁽²⁾ s⁽⁴⁾ s⁽⁴⁾ s⁽¹⁾ s⁽³⁾ . . .

In Table 1, each row indicates a sequence for a different quantity as afunction of index i, which is the column index in the table. The firstrow indicates the q-codewords |ψ_(i)

at input, the second row specifies the sequence of intentional quantumerrors S_(i), the third row specifies the corresponding correctableerror operator Q^((i))∈G_(n), with the syndrome S^((i)), which is thenindicated in the fourth row as measured syndrome Ŝ_(i).

TABLE 2 c-information −1 + 1 +1 − 1 +1 + 1 −1 − 1 +1 + 1 +1 − 1 −1 + 1Q-packet Q⁽²⁾|ψ₁ 

Q⁽⁴⁾|ψ₂ 

Q⁽¹⁾|ψ₃ 

Q⁽³⁾|ψ₄ 

Q⁽¹⁾|ψ₅ 

Q⁽⁴⁾|ψ₆ 

Q⁽²⁾|ψ₇ 

In Table 2 it is shown an example, in which is performed thepiggybacking of 14 classical bits of information−1+1+1−1+1+1−1−1+1+1+1−1−1+1 over a quantum packet composed of 7q-codewords. Each q-codeword |ψ_(i)

=α_(i)|000

+β_(i)|111

is originated by a repetition [[3, 1]] QECC whose error syndromes areS⁽¹⁾=(+1, +1), S⁽²⁾=(−1, +1), S⁽³⁾=(−1, −1), S⁽⁴⁾=(+1, −1). Thus, the 14bits are introduced as error syndromes S⁽²⁾, S⁽⁴⁾, S⁽¹⁾, S⁽³⁾, S⁽¹⁾,S⁽²⁾, shown in the first row as classical information or C-information,determining the Q-packets in the second row of Table 2, introducing thecorresponding intentional errors in the corresponding q-codewords. HereQ^((i))∈Gn is the intentional error having syndrome S^((i)), andtherefore Q⁽¹⁾=no error, Q⁽²⁾=bit-flip on the first qubit, Q⁽³⁾=bit-flipon the second qubit, Q⁽⁴⁾=bit-flip on the third qubit.

In Table 3 it is instead shown how a pattern of errors can be added inpiggyback for frame synchronization. The pattern, which can be on top ofa portion of the user qubits or over an entire packet, must be designedto cope with errors, similarly to classical frame synchronization

TABLE 3 DATA DATA + SYNC WORD . . . |ψ₁ 

|ψ₂ 

|ψ₃ 

|ψ₄ 

Q⁽⁴⁾|ψ₅ 

Q⁽²⁾|ψ₆ 

Q⁽³⁾|ψ₇ 

. . . . . . s⁽¹⁾ s⁽¹⁾ s⁽¹⁾ s⁽¹⁾ s⁽⁴⁾ s⁽²⁾ s⁽³⁾ . . .

As shown in Table 3, quantum information frames, as the one shown in thefirst row of Table 3, are composed of a given number, e.g. 7q-codewords.

A sync word determined by a pattern of syndromes is applied to asequence of q-codewords in the frame. The first four q-codewords haveerrors syndromes attached, S⁽¹⁾, which as shown determine Q⁽¹⁾=no errorat the receiver 13 a, in particular at block 132, thus represent theuser data, or first data, in the frame, while the last three q-codewordsare user data plus a sync word, represented by errors syndrome S⁽⁴⁾,S⁽²⁾, S⁽³⁾ which intentional error P are introduced in the last threecodewords, to form the sync word in the frame. It is underlined that theuser data in the frame relating to the last three q-codewords arepreserved, thus the frame in this portion carries both user data andsync word.

Based on the above, in FIG. 8 it is shown a diagram flow of anembodiment 200 of the method for sending first data as quantuminformation in qubits (|φ

) and second classical data over a quantum channel, in particular inquantum information communication systems like system 10 a or 10 b shownin the following, which includes applying QECC encoding, e.g. by blockto said qubits (|φ

) obtaining quantum information codewords, applied to a noiselesschannel 12 a.

Such method 200 includes a step 210 of applying intentional errors P_(i)having error syndromes S_(i), representing the second classical data tothe quantum information codewords |ψ

obtaining quantum information codewords with intentional errors P_(i)applied upon P_(i)|ψ_(i)

i.e. by using block 112, and

-   -   transmitting 220 from a transmitting side, e.g. module 11 a,        such quantum information codewords with intentional errors        applied upon P_(i)|ψ_(i)        over said quantum channel 12 a which outputs received codewords        P_(i)|ψ_(i)        at a receiving side, in particular at a receiving module 13,    -   computing 230 error syndromes Ŝ_(i), i.e. measuring error        syndromes Ŝ_(i) from said received codewords P_(i)|ψ_(i)        , e.g. at block 131,    -   performing a QECC error correction operation 250, e.g. in block        132, on said received codewords P_(i)|ψ_(i)        by applying a correction operator {circumflex over (P)}_(i)        ^(†)obtained at least by said computed syndromes Ŝ_(i) to obtain        corrected codewords |{circumflex over (ψ)}_(i)        ,    -   outputting 260 such corrected codewords |{circumflex over        (ψ)}_(i)        ) and said computed syndromes Ŝ_(i).

Now, the method in case of a noisy quantum channel will be describedwith reference to the block schematics in FIG. 4, showing a furtherembodiment of a quantum information communication system 10 b.

Also here the intentional error P_(i)∈{Q⁽¹⁾, Q⁽²⁾, . . . , Q^((m))} isapplied on the i-th q-codeword, but the quantum channel 13 b is noisy,i.e. introduces an error E_(i)∈G_(n). If the quantum channel introducesan error E_(i)∈G_(n), the measured syndrome Ŝ_(i) at the receiver side,which is represented by a receiver module 13 b, is that of a combinederror E_(i)P_(i) of the channel error E_(i) and of the intentionalerror, that it is denoted by Ŝ_(i)=S(E_(i)P_(i)) . Therefore, the PSC20, which, as it will be shown in the following, includes blocks 112,113, 12 b, 131, 133, 134, can be seen as a noisy channel, which takes asinput the syndrome S_(i) and produces the output measured syndromeŜ_(i)which can be different from error syndrome S_(i) on the transmitterside 11 b if the quantum channel 12 b introduces a channel error E_(i)at time i, i.e. the time corresponding to the sending of the i-th qubitor q-codeword over the quantum channel 12 b.

To cope with the noisy PSC it is possible to apply classical errorcorrection for the piggy-backed channel, as depicted in FIG. 4.

In FIG. 4, the system 10 b is described therefore, where a transmittermodule 11 includes a classic encoder 113 which encodes classicinformation data B_(k) and outputs error syndromes S_(i), formingc-codewords of a classic code.

Then the controlled error insertion block 112 is configured to apply anintentional error P_(i)∈{Q⁽¹⁾, Q⁽²⁾, . . . , Q^((m))} to said i-thq-codeword |ψ_(i)

based on the error syndromes S_(i) supplied by the classic encoder 113.Quantum information codewords with intentional errors applied upon, i.e.piggybacked, P_(i)|ψ_(i)

are consequently sent over the noisy quantum channel 13 b.

The codes used by the classic encoder 113 to protect the syndromes canbe any classical error correction code type, such as e.g. BCH(Bose-Chaudhuri-Hocquenghem), RS (Reed-Solomon), Convolutional, LDPC(Low Density Parity Check), Turbo, Polar. At the receiver side, i.e. ata receiver module 13 b, the errors introduced by the quantum channel 12b on the received syndromes can be corrected with high probability.

At the receiver side 13 b received codewords E_(i)P_(i)|ψ_(i)

are sent to the QECC correction block 132 and to the syndromecomputation block 131. The syndrome computation block 131 computes inthis case channel affected error syndromes R_(i), which are determinedfrom the received codewords E_(i)P_(i)|ψ_(i)

). Channel affected error syndromes R_(i) are fed to a classicalsyndrome error correction block 133 which is configured to correct theerrors introduced by the noisy quantum channel 12 b in the channelaffected error syndromes R_(i), thus configured to compute from thechannel affected error syndromes R_(i) the measured error syndromesŜ_(i), which are sent both to a block 134 performing error computation,i.e. configured to compute the intentional error syndrome S(P_(i)) andthe channel error syndrome S(E_(i)), which are then fed to thecorrection block 131 outputting the corrected codeword |{circumflex over(ψ)}_(i)

as further output of the communication system corrected in block 14 byapplying the inverse operator of the combined intentional and channelerror operator, {circumflex over (P)}_(i) ^(†)Ê_(i) ^(†).

This because the presence of both intentional errors P_(i) and channelerrors E_(i) requires also to modify the quantum error correctionprocedure at the receiver side. In fact, assuming the channel errorE_(i) introduced by the noisy quantum channel 12 b is correctable, thereare two possibilities:

1) it can happen that the combined error E_(i)P_(i) is correctable, andin this case the quantum correction based on the syndrome of thecombined error E_(i)P_(i) will work;

2) it can happen that the combined error E_(i)P_(i) is not correctable,i.e., that combined error E_(i)P_(i)∉{Q⁽¹⁾, Q⁽²⁾, . . . , Q^((m))}. Forexample, if the correction block 131 is able to correct at most a singlequbit error per q-codeword, the intentional error E_(i) plus an eventualquantum channel error P_(i) could produce errors on two qubits, thatmakes the usual quantum error correction based on the syndrome of thecombined error E_(i)P_(i), i.e. measured syndrome S(E_(i)P_(i)), tofail.

However, this does not represent a problem if classical correction oferrors on the PSC 20 has been successful, since, once the intentionalerror P_(i) has been estimated, the channel error E_(i)∈{Q⁽¹⁾, Q⁽²⁾, . .. , Q^((m))} producing the measured syndrome S(E_(i)P_(i)), is easilyfound.

More precisely, S(E_(i)P_(i))=S(E_(i))°S(P_(i)), where ° denotes theelement-wise product (Hadamard product). Then, since the syndromeelements are ±1, it follows that the error channel syndrome S(E_(i)) isequal to S(E_(i)P_(i))°S(P_(i)).

The block 134 performing error computation in FIG. 4 sends therefore tothe quantum error correction block 131 both the measured syndromeŜ_(i)=S({circumflex over (P)}_(i)), from block and R_(i)°Ŝ_(i)=S(Ê_(i))R_(i) indicates the combined syndrome calculated by the syndrome errorcorrection block 133 on the received state or codeword E_(i)P_(i)|ψ_(i)

. The composite or intentional plus channel error is finally correctedin block 131 by applying the operator {circumflex over (P)}_(i)^(†)Ê_(i) ^(†).

Thus, with this method the original error correction capability of theQECC is not affected by the classical piggyback channel, as long aserrors on the PSC 20 are corrected by classical error correcting codes.

Based on the above, in FIG. 9 it is shown a diagram flow of anembodiment 300 of the method for sending first data as quantuminformation in qubits (|φ

) and second classical data over a quantum channel, in particular inquantum information communication systems like system 10 b, whichincludes applying QECC encoding, e.g. by block to said qubits (|φ

) obtaining quantum information codewords, applied to a noisy channel 12b.

The method 300 includes, prior a step 310, analogous to step 210, ofapplying error syndromes S_(i) representing said second data to thequantum information codewords |ψ

determined by the QECC encoder 11, e.g. by obtaining quantum informationcodewords with intentional errors applied upon, e.g. by using block 112,a step 305 of encoding information data B_(k) with a classical errorcode correction encoder to obtain encoded error syndromes, formingc-codewords of a classical code, which are applied to the quantuminformation codewords |ψ

in step 310, analogous to step 210 besides the fact that error syndromesS_(i) are classically encoded, i.e. determining quantum informationcodewords with intentional errors P_(i) applied upon, P_(i)|ψ_(i)

.

Then the step of transmitting 320 said quantum information codewordswith intentional errors P_(ii) applied upon over said noisy quantumchannel 12 b which outputs received codewords E_(i)P_(i)|ψ_(i)

at the receiving side 13 b is performed.

Then the step 330 of computing error syndromes S_(i;); R_(i) from saidreceived codewords ( E_(i)P_(i)|ψ_(i)

) is performed which in this case includes

-   -   a step 333 of computing channel affected syndromes R_(i) from        the received codewords E_(i)P_(i)|ψ_(i)        , e.g. in block 131 and    -   a step 337 of classical syndrome error correction on said        channel affected syndromes R_(i) to obtain classical corrected        error syndromes S_(i), e.g performed in block 133.

The channel affected syndromes R_(i) are thus corrected with respect toerrors introduced by the noisy quantum channel 12 b by a classicalchannel decoder, block 133.

Then, as shown in FIG. 4, the operation of applying a correctionoperator obtained at least by said computed syndromes R_(i) to obtaincorrected codewords ↑{circumflex over (ψ)}_(i)

includes some more steps as the computed syndromes are the noisy channelaffected syndromes, thus syndromes of the intentional error which carrythe classical are still to be extracted.

Thus, obtaining said correction operator, which is in the end{circumflex over (P)}_(i) ^(†)Ê_(i) ^(†), includes

-   -   performing an operation 340 of computation of the intentional        error {circumflex over (P)}_(i), to which corresponds the        intentional error syndrome S({circumflex over (P)}_(i)) and of        the channel error Ê_(i), to which corresponds the error channel        syndrome S(Ê_(i)) on the basis of said channel affected        syndromes R_(i), obtained at step 333, and said classical        corrected error syndromes S_(i), obtained at step 337,    -   using then said computed intentional error S({circumflex over        (P)}_(i)) and channel error S(Ê_(i)) in step 350, e.g. in block        132, to obtain said correction operator {circumflex over        (P)}_(i) ^(†)Ê_(i) ^(†), in particular as inverse of the        computed intentional error {circumflex over (P)}_(i) and channel        error Ê_(i),    -   finally outputting 360 said corrected codewords and said        classical corrected error syndromes S_(i).

The probability that classical piggybacking causes a not correctableerror on the noisy quantum channel 13 b is thus upper bounded by theresidual syndrome error probability on the PSC after classical errorcorrection, that it is indicated by P_(e,PSC). In other words,piggybacking a classical channel on a quantum channel does not affectthe error correction capability of the QECC correction, with probabilityat least 1−P_(e,PSC).

It is observed that the correction of a quantum error is made with adelay due to the need to wait the correction performed by the classicaldecoder. In this regard, the classical codes for the PSC should bedesigned in order to have a suitable latency.

In FIG. 5 it is shown an example of encoder for the [[3, 1]] QECC, whichmay be an embodiment of QECC encoder 111. The horizontal lines representqubit states as a function of time. A vertical line connecting qubitswith a closed dot on one side and an open dot with cross on the otherside represents a conditional NOT gate (CNOT).

It includes data qubits |φ

as input and the q-codeword |ψ

is composed by the three (n=3) output qubits. The data qubit |φ

=α|0

+β|1

is mapped into the q-codeword α|000

+β|111

using ancilla qubits |α₁

and |α₂

, which are coupled to the data qubit |φ

using CNOT gates, in a way which is known per se.

In FIG. 6 it is shown a circuit for measuring a single qubit |q

according to the operator M with eigenvalues ±1, which can be used toimplement syndrome computation 132. The bottom is an ancilla qubitinitialized in zero state |0

used for measurement. The boxed H represent a Hadamard gate. The outputof the measurement block is ±1.

More in detail a Hadamard gate is performed on the ancilla qubit, whichis then used with operator M to measure the single qubit |q

. Then, a further Hadamard gate is performed on the ancillary qubit.Finally, the state of the ancillary qubit is measured by the block F toextract the error syndrome.

In FIG. 7 it is shown a quantum circuit to apply unitary operator U on aqubit |q

controlled by a classical bit c, which can represent the correctionblock 132, since to obtain the error it is necessary to measureaccording to single qubit operators M with eigenvalues ±1, like X, Y, Z,as shown in FIG. 6. To correct errors it is necessary to apply a unitaryoperator U, like X, Y, Z, controlled by a classical bit c.

In the circuit of FIG. 7, when the control bit is set the output isqubit U|q

, otherwise the qubit |q

is left unchanged. U is a unitary operator.

The solution here described thus substantially provides piggybackingframes of quantum information via intentional errors for both noiselessand noisy quantum channels.

The solution here described adds classical information, i.e. controldata, on top of quantum information, allowing for instance to annotatequbits to describe what they represents, who produced them, etc. Alsothe solution described allows reading and rewriting this annotationwithout destroying the quantum information. Another example ofapplication is related to the possibility to have a quantum network(QN), where nodes exchange quantum information organized in packets.

It will be otherwise understood that the various individual implementingoptions exemplified throughout the figures accompanying this descriptionare not necessarily intended to be adopted in the same combinationsexemplified in the figures. One or more embodiments may thus adopt these(otherwise non-mandatory) options individually and/or in differentcombinations with respect to the combination exemplified in theaccompanying figures.

Without prejudice to the underlying principles, the details andembodiments may vary, even significantly, with respect to what has beendescribed by way of example only, without departing from the extent ofprotection. The extent of protection is defined by the annexed claims.

1. Method for sending first data as quantum information in qubits (|φ

) and classical second data (S_(i)) over a quantum channel, inparticular in quantum information communication systems, which includesapplying QECC encoding to said qubits (|φ

) obtaining quantum information codewords (|ψ

) , wherein said method includes applying intentional errors (P_(i))having error syndromes (S_(i)) representing said second classical datato said quantum information codewords (|ψ

) obtaining quantum information codewords with intentional errors(P_(i)) applied upon (P_(i)|ψ_(i)

), and transmitting from a transmitting side said quantum informationcodewords with intentional errors applied upon (P_(i)|ψ_(i)

) over said quantum channel which outputs received codewords(P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

) at a receiving side, computing error syndromes (Ŝ_(i); R_(i)) fromsaid received codewords (P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

), performing a QECC error correction operation on said receivedcodewords (P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

) by applying a correction operator ({circumflex over (P)}_(i) ^(†),{circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)) obtained at least by saidcomputed syndromes (Ŝ_(i); R_(i)) to obtain corrected codewords(|{circumflex over (ψ)}_(i)

), outputting said corrected codewords (|{circumflex over (ψ)}_(i)

) and said computed syndromes (Ŝ_(i)).
 2. The method of claim 1, whereinincludes encoding information data (B_(k)) with a classical error codecorrection encoder to obtain encoded error syndromes (S_(i)) which areapplied as said error syndromes to the quantum information codewords (|ψ

), said computing error syndromes (S_(i;); R_(i)) from said receivedcodewords (E_(i)P_(i)|ψ_(i)

) including a step of computing channel affected syndromes (R_(i)) fromthe received codewords (E_(i)P_(i)|ψ_(i)

) and a step of classical syndrome error correction on said channelaffected syndromes (R_(i)) to obtain classical corrected error syndromes(S_(i)), said applying a correction operator ({circumflex over (P)}_(i)^(†), {circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)) obtained at least bysaid computed syndromes (S_(i;); R_(i)) to obtain corrected codewords(|{circumflex over (ψ)}_(i)

) includes obtaining said correction operator ({circumflex over (P)}_(i)^(†), {circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)) by performing anoperation of computation of the intentional error ({circumflex over(P)}_(i)) and of the channel error to which is associated the syndrome(S(Ê_(i))) on the basis of said channel affected syndromes (R_(i)) andsaid classical corrected error syndromes (S_(i)), using said computedintentional error ({circumflex over (P)}_(i)) and channel error (Ê_(i))to obtain said correction operator ({circumflex over (P)}_(i) ^(†),{circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)), in particular as inverse ofthe computed intentional error ({circumflex over (P)}_(i)) and channelerror S(Ê_(i)).
 3. The method of claim 1, wherein said applying acorrection operator ({circumflex over (P)}_(i) ^(†), {circumflex over(P)}_(i) ^(†)Ê_(i) ^(†)) obtained at least by said computed syndromes(S_(i;); R_(i)) to obtain corrected codewords (|{circumflex over(ψ)}_(i)

) includes obtaining the intentional error from said computed syndromesand computing the correction on the basis of said intentional error, inparticular as inverse of the intentional error ({circumflex over(P)}_(i) ^(†)).
 4. The method of claim 1, wherein applying intentionalerrors (P_(i)) having error syndromes (S_(i)) representing said secondclassical data to said quantum information codewords (|ψ

) obtaining quantum information codewords with intentional errors(P_(i)) applied upon (P_(i)|ψ_(i)

) includes introducing in the quantum information codewords intentionalerrors (P_(i)) determined by corresponding syndromes (S_(i)).
 5. Themethod of claim 1, wherein said second classical data (S_(i)) representcommunication control data.
 6. The method of claim 1, wherein saidsecond classical data (S_(i)) represent a synchronization word which isattached to selected codewords.
 7. A quantum communication systemconfigured to send first data as quantum information in qubits (|φ

) and classical second data (S_(i)) over a quantum channel, comprising aquantum information transmission module which includes a QECC encoderconfigured to apply QECC encoding to said qubits (|φ

) obtaining quantum information codewords (|ψ

), wherein said quantum information transmission module is configured toapply (210) intentional errors (P_(i)) having error syndromes (S_(i))representing said second classical data to said quantum informationcodewords (|ψ

) obtaining quantum information codewords with intentional errors(P_(i)) applied upon (P_(i)|φ_(i)

), and transmit from a transmitting side said quantum informationcodewords with intentional errors applied upon (P_(i)|ψ_(i)

) over said quantum channel which outputs received codewords(P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

) at a receiver module comprised in say system, said receiver modulebeing configured to compute error syndromes (S_(i;); R_(i)) from saidreceived codewords (P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

); perform a QECC error correction operation on said received codewords(P_(i)|ψ_(i)

; E_(i)P_(i)|ψ_(i)

) by applying a correction operator ({circumflex over (P)}_(i) ^(†),{circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)) obtained at least by saidcomputed syndromes (S_(i;); R_(i)) to obtain corrected codewords(|{circumflex over (ψ)}_(i)

), output said corrected codewords (|{circumflex over (ψ)}_(i)

) and said extracted syndromes (S_(i;)).
 8. The system of claim 7,wherein said transmitter module includes a classical error codecorrection encoder configured to encode information data (B_(k)) toobtain encoded error syndromes (S_(i)) which are applied as said errorsyndromes to the quantum information codewords (|ψ

), said receiver module configured to compute error syndromes (S_(i;);R_(i)) from said received codewords (E_(i)P_(i)|ψ_(i)

) including one or more modules configure to compute channel affectedsyndromes (R_(i)) from the received codewords (E_(i)P_(i)|ψ_(i)

), and perform a classical syndrome error correction on said channelaffected syndromes (R_(i)) to obtain classical corrected error syndromes(S_(i)), an application of a correction operator ({circumflex over(P)}_(i) ^(†), {circumflex over (P)}_(i) ^(†)Ê_(i) ^(†)) obtained atleast by said computed syndromes (S_(i;); R_(i)) to obtain correctedcodewords (|{circumflex over (ψ)}_(i)

) including obtaining said correction operator ({circumflex over(P)}_(i) ^(†), {circumflex over (P)}_(i) ^(†), Ê_(i) ^(†)) by performingan operation of computation of the intentional error ({circumflex over(P)}_(i)) and of the channel error (Ê_(i)) on the basis of said channelaffected syndromes (R_(i)) and said classical corrected error syndromes(S_(i)), using said computed intentional error ({circumflex over(P)}_(i)) and channel error (Ê_(i)) to obtain said correction operator({circumflex over (P)}_(i) ^(†), {circumflex over (P)}_(i) ^(†)Ê_(i)^(†)), in particular as inverse of the computed intentional error({circumflex over (P)}_(i) and channel error Ê_(i).
 9. The system ofclaim 7, wherein the receiver module is configured to apply a correctionoperator ({circumflex over (P)}_(i) ^(†), {circumflex over (P)}_(i)^(†)Ê_(i) ^(†)) obtained at least by said computed syndromes (S_(i;);R_(i)) to obtain corrected codewords (|{circumflex over (ψ)}_(i)

includes obtaining the intentional error from said computed syndromesand computing the correction on the basis of said intentional error, inparticular as inverse of the intentional error ({circumflex over(P)}_(i) ^(†)).